[Math Lair] Egyptian Fractions

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Two thousand years before Christ, the Babylonians used decimals (sexagesimals, actually) to represent fractions. The Babylonian base 60 system was handy for representing many different fractions since 60 divides 2, 3, 4, 5, and 6, among other numbers (see also shortcuts for checking for divisibility). The Egyptians, on the other hand, had a clumsier system for expressing fractions. While they understood rational fractions with numerators greater than one, they had no symbols for them. With the exception of ⅔ (two-thirds), for which the Egyptians had a special symbol (literally "one over one and a half"), they had symbols only for unit fractions. Unit fractions are fractions whose numerator is 1; they are the reciprocals of natural numbers. Examples of unit fractions are ½, 1/3, 1/5, 1/192,754, and so on.

To work with non-unit fractions, the Egyptians expressed such fractions as sums of distinct unit fractions. So, ¾ would be represented as ½ + ¼. This can become cumbersome, so the Ancient Egyptians used tables. For example, the Rhind papyrus contains a table in which every fraction of the form 2/b is expressed as a sum of distinct unit fractions, where b is an odd integer between 5 and 101.

Interestingly, although the Egyptian system is much more complicated than the Babylonian system, or our modern system of having fractions with any numerator and denominator (which the ancient Chinese were also able to handle), the ancient Greeks and the Romans used this unit fraction system, although they also represented fractions in other ways as well. As a matter of fact, this system of unit fractions survived in Europe until the 17th century.

An interesting mathematical recreation is to determine the "best" representation of a fraction in Egyptian fractions. There are several meanings of "best". For example, it could mean minimizing the number of terms, or minimizing the largest denominator, or minimizing the sum of the denominators, or some other criterion or criteria. It is obvious that any proper fraction can be expressed as the sum of unit fractions if a repetition of terms is allowed. For example, 3/7 = 1/7 + 1/7 + 1/7. This isn't allowed in Egyptian fractions; all of the fractions in an expansion must have different denominators.

A famous algorithm for writing any proper fraction as the sum of a finite number of distinct Egyptian fractions was first published in 1202 by Fibonacci in his book Liber Abaci. This algorithm, which is a "greedy algorithm", is fairly simple. Find the largest unit fraction not greater than the proper fraction that you want to find an expansion for. Subtract that unit fraction from the fraction to obtain another proper fraction. The second term of the expansion is the largest unit fraction not greater than that proper fraction. Continue until you obtain a remainder that is a unit fraction. This algorithm always works, and always generates a series of Egyptian fractions containing a number of terms no greater than the value of the numerator.

This algorithm doesn't always generate the "best" expansion, however. For example, the sequence generated by 2/21 is 1/11 + 1/231. A "nicer" expansion, though, is 1/15 + 1/35. To deal with fractions of the form 2/xy, with x not equal to y, the formula 2/xy = 1/(x((x+y)/2)) + 1/(y((x+y)/2)) can be used.

One interesting unsolved problem is: Can a proper fraction 4/b always be expressed as the sum of three or fewer unit fractions? This conjecture has been verified to extremely large values of b, but has not been proven.